On one-sided division infinite-dimensional normed real algebras

Eight-Dimensional Real Quadratic Division Algebras

Given a euclidean vector space V , a linear map η : V ∧ V → V is called dissident in case v, w, η(v∧w) are linearly independent whenever so are v, w ∈ V . The problem of classifying all real quadratic division algebras is reduced to the problem of classifying all eight-dimensional real quadratic division algebras, and further to the problem of classifying all dissident maps η : R ∧ R → R. Shoul...

Riemannian Geometry over different Normed Division Algebras

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkähler and G2-man...

One-sided Projections on C∗-algebras

In [4], the notion of a complete one-sided M -ideal for an operator space X was introduced as a generalization of Alfsen and Effros’ notion of an M -ideal for a Banach space [1]. In particular, various equivalent formulations of complete one-sided M -projections were given. In this paper, some sharper equivalent formulations are given in the special situation that X = A, a C∗-algebra (in which ...

Infinite-dimensional Lie algebras

Notes on normed algebras

All vector spaces and so forth here will be defined over the complex numbers. If z = x+i y is a complex number, where x, y are real numbers, then the complex conjugate of z is denoted z and defined to be x− i y. The complex conjugate of a sum or product of complex numbers is equal to the corresponding sum or product of complex conjugates. The modulus of a complex number z is the nonnegative rea...